We study estimation of piecewise smooth signals over a graph. We propose a $\ell_{2,0}$-norm penalized Graph Trend Filtering (GTF) model to estimate piecewise smooth graph signals that exhibits inhomogeneous levels of smoothness across the nodes. We prove that the proposed GTF model is simultaneously a k-means clustering on the signal over the nodes and a minimum graph cut on the edges of the graph, where the clustering and the cut share the same assignment matrix. We propose two methods to solve the proposed GTF model: a spectral decomposition method and a method based on simulated annealing. In the experiment on synthetic and real-world datasets, we show that the proposed GTF model has a better performances compared with existing approaches on the tasks of denoising, support recovery and semi-supervised classification. We also show that the proposed GTF model can be solved more efficiently than existing models for the dataset with a large edge set.

翻译：我们研究了图上分段平滑信号的估计问题。提出了一种基于$\ell_{2,0}$范数惩罚的图趋势滤波（GTF）模型，用于估计节点间存在不均匀平滑程度的分段平滑图信号。我们证明，所提出的GTF模型同时实现了节点信号的k-means聚类和图上边的最小割，其中聚类和割共享相同的分配矩阵。我们提出了两种求解该GTF模型的方法：谱分解方法和基于模拟退火的方法。在合成数据集和真实世界数据集上的实验中，我们表明，与现有方法相比，所提出的GTF模型在去噪、支持恢复和半监督分类任务上具有更优的性能。我们还表明，对于边集较大的数据集，所提出的GTF模型比现有模型更高效。